244 XIV INTEGRATION IN LOCALLY COMPACT GROUPS If ju is any measure on G, we denote by fl the image of fi under the map- ping xt-+x~1ofG onto G, so that we have (14.1.4) </,£> = </,/<> for all /e tfC(G). It follows immediately from the definitions that we have (j(s)/y = 60)/, each side being the function x^/Os""1*"1); and therefore, for any measure /z on G, we have (y(s)/z)" = 8(s)fi. Hence if /z is a left- invariant measure, # is a right-invariant measure, and vice versa. (14.1.5) Let G be a locally compact group. Then there exists a nonzero left- invariant positive measure n on G, and every other left-invariant measure on G is of the form a/i, where aeC. (1) Existence. Let <#** denote the set of functions g e Jf R(G) which are ^0 and not the zero function. For each/e «#*R(G) and each #e jf* , there exist positive real numbers cl9..., cr and elements sly..., sr in G such that (14.15.1) /^j>iY(*)0 r (i.e., such that/(*)^ ]T cig(s^ix) for all xeG). For there exists a non- i = l empty open subset U in G such that a = inf g(x) > 0; since Supp(/) is com- jceU pact, there exist a finite number of points st e G (1 ^ / ^ r) such that the jjU cover Supp(/), and then (14.1.5.1) is satisfied by taking q = ||/|| fa for all i. We shall denote by (/: g) the greatest lower bound of the numbers r X Cj, for all systems (q,..., cr, slt..., sr) satisfying (14.1.5.1). The symbol i=l (/: g) has the following properties: (i) (Y(*)/:<7) = (/:<?) for all /e ^fR(G), getfl, seG; (ii) (af:g) = a(f:g) for all /eJfR(G), geJtT*+> a^O; for all A,f2eJl (iv) (f:g)^supf(x)/SMpg(x) for all /e JfR(G), getf*; xeG xeG (v) (f:h)^(f:g)(9'K) for all /e JfR(G) and 0,hintf*+; for all /,/0,^ inJT*,uality). (Consider first the case where / is bounded and of compact